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The graded ring of quaternionic modular forms of degree 2

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Abstract

In the 1960's Igusa [I] described the graded ring of Siegel modular forms of degree 2. In the sequel a few more concrete examples of graded rings of modular forms with respect to paramodular groups or Hermitian modular groups were given, e.g. by Freitag [F], Aoki and Ibukiyama [AI] or Dern and Krieg [DK].

Moreover Freitag and Hermann [FH] investigated some modular varieties of low dimension. In particular they discussed the modular variety where denotes the extended modular group of degree 2 over the Hurwitz quaternions. They described it as a major problem to determine the graded ring of modular forms in terms of generators and relations.

In this paper we determine this graded ring completely. Surprisingly it turns out to be a polynomial ring in 7 algebraically independent indeterminates given by Siegel-Eisenstein series of weight up to 24. The proof consists of applications of the results by Freitag and Hermann [FH] and the results on Hermitian modular forms with respect to in [DK]. Additionally we have to put more emphasis on the commutator subgroup of the modular group calculated in [KW] and have to find new descriptions of the Borcherds products constructed in [FH]. Finally we consider the theta series from [FH]. We show that 6 generators can also be chosen as polynomials in these theta series, where one additionally needs the Siegel-Eisenstein series of weight 6.

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  1. Lehrstuhl A für Mathematik, RWTH Aachen, 52056, Aachen, Germany

    A. Krieg

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  1. A. Krieg
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Krieg, A. The graded ring of quaternionic modular forms of degree 2. Math. Z. 251, 929–942 (2005). https://doi.org/10.1007/s00209-005-0840-7

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  • DOI: https://doi.org/10.1007/s00209-005-0840-7

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